To solve the given problem, we need to calculate the reverberation time (RT)
using the Sabine formula and determine the number of absorbing units
(Sabins) for different scenarios. We'll follow these steps:
𝑉=length×width×heightV=length×width×height 𝑉=40 m×30 m×15 m=18,000
1. Calculate the volume of the auditorium:
m3V=40m×30m×15m=18,000m3
2. Calculate the total surface area:
Total surface area=2(length×height+width×height+length×width)Total surface a
rea=2(length×height+width×height+length×width) Total surface area=2(40
m×15 m+30 m×15 m+40 m×30 m)Total surface area=2(40m×15m+30m×15m+
40m×30m) Total surface area=2(600+450+1200)
m2Total surface area=2(600+450+1200)m2 Total surface area=2(2250)
m2Total surface area=2(2250)m2 Total surface area=4500
m2Total surface area=4500m2
3. Identify the areas covered by different materials and their
respective absorption coefficients (α) at a mid-frequency (let’s
assume a typical value, if not provided):
Cement plaster walls: 1800 m21800m2, 𝛼=0.02α=0.02
Concrete floor: 1200 m21200m2, 𝛼=0.01α=0.01
Acoustical material ceiling: 1200 m21200m2, 𝛼=0.60α=0.60
Curtains on walls: 330 m2330m2, 𝛼=0.35α=0.35
𝐴=∑(surface area×𝛼)A=∑(surface area×α) 𝐴cement plaster=1800
4. Calculate the total absorption in Sabins for each material:
𝐴concrete floor=1200 m2×0.01=12 SabinsAconcrete floor=1200m2×0.01=12Sabins
m2×0.02=36 SabinsAcement plaster=1800m2×0.02=36Sabins
𝐴acoustical ceiling=1200 m2×0.60=720 SabinsAacoustical ceiling
=1200m2×0.60=720Sabins 𝐴curtains=330 m2×0.35=115.5 SabinsAcurtains
=330m2×0.35=115.5Sabins
Total absorption without audience=36+12+720+115.5=883.5 SabinsTotal absor
ption without audience=36+12+720+115.5=883.5Sabins
𝑅𝑇=0.161×𝑉𝐴RT=A0.161×V 𝑅𝑇=0.161×18,000 m3883.5
5. Calculate the reverberation time (RT) using the Sabine formula:
SabinsRT=883.5Sabins0.161×18,000m3 𝑅𝑇≈3.28 secondsRT≈3.28seconds
6. Calculate the absorption added by the audience:
𝛼audience=0.45
Sabins/personαaudience=0.45Sabins/person 𝐴audience=2000×0.45=900
Full audience: 2000 seats, typical absorption per person:
SabinsAaudience=2000×0.45=900Sabins
7. Recalculate the total absorption for different audience scenarios:
𝐴total, no audience=883.5 SabinsAtotal, no audience=883.5Sabins
𝑅𝑇no audience=0.161×18,000883.5≈3.28 secondsRTno audience=883.50.161×18,000
a) No audience:
≈3.28seconds
𝐴audience, one-third=20003×0.45=300
SabinsAaudience, one-third=32000×0.45=300Sabins 𝐴total, one-
b) Audience factor is one-third:
𝑅𝑇one-third=0.161×18,0001183.5≈2.45 secondsRTone-third=1183.50.161×18,000
third=883.5+300=1183.5 SabinsAtotal, one-third=883.5+300=1183.5Sabins
≈2.45seconds
c) Full audience: 𝐴total, full audience=883.5+900=1783.5
𝑅𝑇full audience=0.161×18,0001783.5≈1.62 secondsRTfull audience
SabinsAtotal, full audience=883.5+900=1783.5Sabins
=1783.50.161×18,000≈1.62seconds
8. Calculate the extra absorbing units needed to achieve the optimal
RT of 1.2 seconds:
𝑅𝑇optimum=1.2 secondsRToptimum=1.2seconds
𝐴required=0.161×18,000𝑅𝑇optimumArequired=RToptimum0.161×18,000
required=0.161×18,0001.2≈2415 SabinsArequired=1.20.161×18,000≈2415Sabins
Extra absorbing units needed:Extra absorbing units needed:
No audience: 𝐴extra, no audience=2415−883.5=1531.5 SabinsAextra, no audience
=2415−883.5=1531.5Sabins
Audience factor one-third: 𝐴extra, one-third=2415−1183.5=1231.5
SabinsAextra, one-third=2415−1183.5=1231.5Sabins
Full audience: 𝐴extra, full audience=2415−1783.5=631.5
SabinsAextra, full audience=2415−1783.5=631.5Sabins
In summary, the auditorium requires additional absorbing units to achieve
the optimal reverberation time of 1.2 seconds, depending on the presence
and number of the audience.
